Mathematica Bohemica, Vol. 142, No. 3, pp. 243-262, 2017

Existence of infinitely many weak solutions for some quasilinear $\vec{p}(x)$-elliptic Neumann problems

Ahmed Ahmed, Taghi Ahmedatt, Hassane Hjiaj, Abdelfattah Touzani

Received July 14, 2015.  First published January 2, 2017.

Abstract:  We consider the following quasilinear Neumann boundary-value problem of the type $ - \displaystyle\sum_{i=1}^N\frac{\partial}{\partial x_i}a_i\Big(x,\frac{\partial u}{\partial x_i}\Big) + b(x)|u|^{p_0(x)-2}u = f(x,u)+ g(x,u) &\text{in} \Omega, \quad\dfrac{\partial u}{\partial\gamma} = 0 &\text{on} \partial\Omega. $ We prove the existence of infinitely many weak solutions for our equation in the anisotropic variable exponent Sobolev spaces and we give some examples.
Keywords:  Neumann problem; quasilinear elliptic equation; weak solution; variational principle; anisotropic variable exponent Sobolev space
Classification MSC:  35J20, 35J62
DOI:  10.21136/MB.2017.0037-15

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Affiliations:   Ahmed Ahmed, Taghi Ahmedatt, Abdelfattah Touzani, Department of Mathematics, Laboratory LAMA, Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdellah, BP 1796 Atlas, Fez, Morocco, e-mail:,,; Hassane Hjiaj, Department of Mathematics, Faculty of Sciences, Tetouan University Abdelmalek Essaadi, Quartier M'haneche II, Avenue Palestine, BP 2121, Tetouan 93000, Morocco, e-mail:

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